I am curious, If a Amritdhari Gursikh family adopted an non-Punjabi, non-Sikh older child (especially a boy) who is between the ages of 7 to 10. Should the adoptive family permit the child to cut hairs?

Sikhi is the Dharm of Akal Purakh itself. Which means Akal Purakh doesnt like hair being cut. A true Amritdhari will never cut his/her hair ever, nor of anyone else's. As simple as that.